Nilpotent Decomposition of Solvable Lie Algebras∗ Liqun Qi† September 11, 2019 Abstract Semisimple Lie algebras have been completely classified by Cartan and Killing. The Levi theorem states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its largest solvable ideal and a semisimple Lie algebra. These focus the classification of solvable Lie algebras as one of the main chal- lenges of Lie algebra research. One approach towards this task is to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones. In this paper, we propose another approach, i.e., to decompose a solvable nonnilpotent Lie algebra to two nilpotent Lie algebras which are called the left and right nilpotent algebras of the solvable algebra. The right nilpotent algebra is the smallest ideal of the lower central series of the solvable algebra, while the left nilpotent algebra is the factor algebra of the solvable algebra and its right nilpotent algebra. We show that the solvable algebras are decomposable if its left nilpotent algebra is an Abelian algebra of dimension higher than one and its arXiv:1903.03776v9 [math.RA] 10 Sep 2019 right algebra is an Abelian algebra of dimension one. We further show that all the solvable algebras are isomorphic if their left nilpotent algebras are Heisen- berg algebras of fixed dimension and their right algebras are Abelian algebras of dimension one. Key words. Lie algebra, solvable Lie algebra, nilpotent Lie algebra, Abelian algebras, Heisenberg algebras. AMS subject classifications. 15A99, 17B66 ∗This research was supported by the Hong Kong Research Grant Council (Grant No. PolyU 15300715, 15301716 and 15300717). †Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong; ([email protected]). 1 1 Introduction Lie algebra is a branch of algebra, closely related with the Lie group theory, has various applications in physics and mechanics [5, 15]. Sophus Lie (1842-1899) started the research of Lie group and Lie algebra [8]. Later, Friedrich Engel (1861-1941), Wilhelm Killing (1847-1923), Elie´ Cartan (1869-1951), Hermann Weyl (1885-1955), Bartel van der Waerden (1903-1996), Nathan Jacobson (1910-1999) and Eugene Dynkin (1924- 2014) made various contributions to the developments of Lie algebra [3, 4, 5, 6, 15]. In particular, Cartan [2] characterized solvable Lie algebras and semisimple algebras by their Killing forms and completely classified semisimple Lie algebras. Eugenio Elia Levi [7] showed that a finite-dimensional Lie algebra is a semidirect sum of a solvable ideal and a semisimple Lie subalgebra. As semisimple Lie algebras were completely classified, people turn their attentions from the classification of general finite-dimensional Lie algebras to the classification of solvable Lie algebras and its subclass nilpotent Lie algebras [4, 11, 15]. The problem is completely solved up to dimension 7, and the difficulty increases rapidly as the dimension increases [15]. The 1996 book of Goze and Khakimdjanov [4] devoted to the study of nilpotent Lie algebras. Some classes of nilpotent Lie algebras, such as Abelian Lie algebras, Heisen- berg algebras, two-step algebras, filiform Lie algebras, characteristically nilpotent Lie algebras, nilpotent standard algebras, were studied in [4]. How can solvable nonnilpotent Lie algebras be classified? Starting from 1993 [12], Winternitz and his collaborators pursued such an approach: to take a class of nilpotent Lie algebras and construct all extensions of these algebras to solvable ones [9, 13, 14]. Also see [16]. These were further put together in the 2014 book of Snobl˘ and Winternitz [15, Part 3]. For a finite dimensional Lie algebra, both the largest solvable ideal and the largest nilpotent ideal exist. They are called the radical and the nilradical of the Lie algebra. The Levi decomposition states that every finite dimensional Lie algebra is isomorphic to a semidirect sum of its radical and a semisimple Lie algebra. The approach of Snobl˘ and Winternitz [15] uses the nilradical of a solvable nonnilpotent Lie algebra as the starting point to study such a solvable Lie algebra. A rational algorithm for identifying the nilradical of a Lie algebra was presented in [15, Chapter 7]. As discussed in [3, Chapter 3], even in the three dimensional case, if the dimension of the derived algebra is two, then there are infinitely many non-isomorphic Lie alge- bras. This example will be discussed as Example 2 in Section 3 of this paper. They are solvable nonnilpotent Lie algebras. Therefore, the classification work of solvable nonnilpotent Lie algebras can only divide such Lie algebras to classes. Some classes may contain infinitely many non-isomorphic Lie algebras. In this paper, we propose to decompose a finite dimensional solvable nonnilpotent 2 Lie algebra to two nilpotent Lie algebras. The starting point of such an approach is the smallest ideal of the lower central series of a solvable nonnilpotent Lie algebra. In [10], we studied three minimax ideal relations of a Lie algebra. The smallest ideal of the lower central series is at one of the three minimax places of a Lie algebra. We introduced near perfect ideas for a finite dimensional Lie algebra in [10]. The largest near perfect ideal of a finite dimensional Lie algebra always exists and is exactly the smallest ideal of the lower central series of that Lie algebra. We call it the near perfect radical of that Lie algebra. A Lie algebra is nilpotent if and only if its near perfect radical is zero. The factor algebra of a Lie algebra by its near perfect radical is always nilpotent. For a solvable Lie algebra, its near perfect radical is nilpotent. Using these facts, we decompose a finite dimensional solvable nonnilpotent Lie algebra to two nilpotent Lie algebras. The first nilpotent Lie algebra is the factor algebra of the solvable Lie algebra by its near perfect radical. We call it the left nilpotent algebra of that solvable Lie algebra. The second Lie algebra is the near perfect radical of that solvable Lie algebra. We call it the right nilpotent algebra of that solvable Lie algebra. Then we may classify solvable Lie algebras by such a nilpotent decomposition, and study solvable Lie algebras with different nilpotent decompositions separately. For example, we may study A(n1)- A(n2) algebras, which are solvable Lie algebras whose left and right nilpotent algebras are both Abelian and with dimensions n1 and n2 respectively. Actually, the simplest solvable nonnilpotent Lie algebra is the A(1)-A(1) algebra, which is of dimension 2. See Example 1 in Section 3. On the other hand, the three dimensional case stated in the last paragraph is actually the A(1)-A(2) algebra, which will be studied as Example 2 in Section 3. We also combine the lower central series of the left and right nilpotent algebras of a solvable nonnilpotent Lie algebra together, and call it the extended lower central series of the solvable Lie algebra. This strengthens the lower central series of such a solvable Lie algebra. In the next section, some preliminaries are given. In Section 3, we introduce the nilpotent decomposition and the extended lower central series for a finite dimensional solvable nonnilpotent Lie algebra, and study such a decomposition for dimension up to 4. We study A(n1)-A(n2) algebras in Section 4. We show that an A(n)-A(1) algebra with n ≥ 2 is the direct sum of an A(1)-A(1) algebra and n − 1 Abelian algebras A(1), and conjecture that an A(n1)-A(n2) algebra with n1 > n2 is always decomposable. Then we give a theorem for the structure of general A(n1)-A(n2) algebras, for further classifying such solvable Lie algebras. In Section 5, we study A(n)-Hp algebras, Hp-A(n) algebras and Hp-Hq algebras. Here, Hp is the pth order Heisenberg algebra, which is of dimension 2p + 1. We show that all the Hp-A(1) algebras are isomorphic. We then give a theorem for the structure 3 of general Hp-A(n) algebras. Thus, we completely classify A(1)-A(1) algebras and Hp-A(1) algebras, and give outlines for classifying A(n1)-A(n2) algebras and Hp-A(n) algebras. Some final remarks are made in Section 6. 2 Preliminaries In this paper, we only study finite dimensional Lie algebras. The related field F is either the field of complex numbers or the field of real numbers. Suppose that L is an n-dimensional Lie algebra defined on F. Denote the Lie bracket operation on L by [·, ·]. Then [·, ·] is a bilinear operation, and for x, y, z ∈ L, we have [x, y]= −[y, x] (2.1) and the Jacobi identity [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0. (2.2) If [x, y] ≡ 0 for any x, y ∈ L, then L is called an Abelian algebra. We denote an Abelian algebra of dimension n by A(n). When n > 1, Abelian algebra A(n) can always be decomposed to a direct sum of n Abelian algebras A(1). In [15, Part 4], A(1) is denoted as n1,1. A subspace I of L is called a subalgebra of L if for any x, y in I,[x, y] ∈ I. If furthermore for any x ∈ L and y ∈ I, we have [x, y] ∈ I, then I is called an ideal of L. The center of L, defined as Z(L)= {x ∈ L :[x, L]=0}, is an ideal of L. Suppose that I and J are two ideals of L. Then I ∩ J, I + J := {x + y : x ∈ I, y ∈ J} and [I,J] := Span{[x, y]: x ∈ I, y ∈ J} are also ideals of L.